where we define \(\hat{\delta }_{\rm LP}(\mu )=\delta _{\rm LP}(\mu )\)  when \(\mu \notin \mathcal {E}\) ,
\(\hat{\delta }_{\rm LP}(\mu )=\Delta _\mu \)  otherwise.
The region \(w \ge \frac{1}{2} s^2 -\hat{\delta }_{\rm LP}(s)\)  is the region above the blue jagged curve and red points
in Figure REF  (the black and purple lines will be discussed momentarily). One can show that the remaining axioms for Bridgeland stability conditions (HN filtration and support property) are indeed satisfied by the pair \(\sigma =(Z,\mathcal {A}(s))\)  in this region. Moreover, the resulting stability condition turns out to exhaust the subset \(U\subset \operatorname{Stab}(\mathcal {C})\)  of geometric stability conditions, defined as those stability conditions for which all skyscraper sheaves \(\mathcal {O}_x\)  with \(x\in {P}^2\)  are \(\sigma \) -stable with the same phase. The connected component of \(\operatorname{Stab}(\mathcal {C})\)  containing \(U\) ,
denoted by \(\operatorname{Stab}^\dagger (\mathcal {C})\) , is the union of the images of (the closure of) \(U\)  under the group \(\Gamma _1(3)\)  of auto-equivalences
of \(\mathcal {C}\)  generated by spherical twists, and is simply connected [1]}. It is unknown whether \(\operatorname{Stab}(\mathcal {C})\)  might have other connected components.
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